Torisphere (EntityTopic, 11)
From Hi.gher. Space
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| - | The toraspherinder is a four-dimensional torus formed by taking an uncapped [[spherinder]] and connecting its ends through its inside | + | The toraspherinder is a [[four-dimensional torus]] formed by taking an uncapped [[spherinder]] and connecting its ends through its inside. Its [[toratopic dual]] is the [[toracubinder]]. |
== Equations == | == Equations == | ||
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*All points (''x'', ''y'', ''z'', ''w'') that lie on the [[surcell]] of a toraspherinder will satisfy the following equation: | *All points (''x'', ''y'', ''z'', ''w'') that lie on the [[surcell]] of a toraspherinder will satisfy the following equation: | ||
| - | <blockquote>(√(''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>) | + | <blockquote>(√(''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>) − ''R'')<sup>2</sup> + ''w''<sup>2</sup> = ''r''<sup>2</sup></blockquote> |
*The parametric equations are: | *The parametric equations are: | ||
Revision as of 10:59, 12 March 2011
The toraspherinder is a four-dimensional torus formed by taking an uncapped spherinder and connecting its ends through its inside. Its toratopic dual is the toracubinder.
Equations
- Variables:
r ⇒ minor radius of the toraspherinder
R ⇒ major radius of the toraspherinder
- All points (x, y, z, w) that lie on the surcell of a toraspherinder will satisfy the following equation:
(√(x2 + y2 + z2) − R)2 + w2 = r2
- The parametric equations are:
x = r cos a cos b cos c + R cos b cos c
y = r cos a cos b sin c + R cos b sin c
z = r cos a sin b + R sin b
w = r sin a
- The hypervolumes of a toraspherinder are given by:
total edge length = 0
total surface area = 0
surcell volume = 8π2Rr2
bulk = 8π2Rr33-1
- The realmic cross-sections (n) of a toraspherinder are:
Unknown
| Notable Tetrashapes | |
| Regular: | pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron |
| Powertopes: | triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate |
| Circular: | glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder |
| Torii: | tiger • torisphere • spheritorus • torinder • ditorus |
| 6a. (II)(II) Duocylinder | 6b. ((II)(II)) Tiger | 7a. (III)I Spherinder | 7b. ((III)I) Toraspherinder | 8a. ((II)I)I Torinder | 8b. (((II)I)I) Ditorus |
| List of toratopes | |||||
